# How do you find the explicit formula and calculate term 20 for 3, 9 , 27, 81, 243?

Jul 17, 2015

The explicit formula for the progression is $\textcolor{red}{{t}_{n} = {3}^{n}}$ and $\textcolor{red}{{t}_{20} = \text{3 486 784 401}}$.

#### Explanation:

This looks like a geometric sequence, so we first find the common ratio $r$ by dividing a term by its preceding term.

Your progression is $3 , 9 , 27 , 81 , 243$.

${t}_{2} / {t}_{1} = \frac{9}{3} = 3$

${t}_{3} / {t}_{2} = \frac{27}{9} = 3$

${t}_{4} / {t}_{3} = \frac{81}{27} = 3$

${t}_{5} / {t}_{4} = \frac{243}{81} = 3$

So $r = 3$.

The ${n}^{\text{th}}$ term in a geometric progression is given by:

${t}_{n} = a {r}^{n - 1}$ where $a$ is the first term and $r$ is the common difference

t_n = ar^(n-1) =3(3)^(n-1) = 3^1 × 3^(n-1) = 3^(n-1+1)
${t}_{n} = {3}^{n}$
If $n = 20$, then
${t}_{20} = {3}^{20} = \text{3 486 784 401}$